| L(s) = 1 | + 54·5-s + 610·7-s + 1.45e3·9-s − 1.06e3·11-s + 9.63e3·17-s + 1.37e4·19-s − 4.12e4·23-s + 1.56e4·25-s + 3.29e4·35-s − 1.42e5·43-s + 7.87e4·45-s − 7.51e4·47-s + 1.17e5·49-s − 5.73e4·55-s + 5.70e4·61-s + 8.89e5·63-s − 3.84e5·73-s − 6.47e5·77-s + 1.59e6·81-s + 2.26e6·83-s + 5.20e5·85-s + 7.40e5·95-s − 1.54e6·99-s + 4.12e6·101-s − 2.22e6·115-s + 5.87e6·119-s + 1.77e6·121-s + ⋯ |
| L(s) = 1 | + 0.431·5-s + 1.77·7-s + 2·9-s − 0.797·11-s + 1.96·17-s + 2·19-s − 3.38·23-s + 25-s + 0.768·35-s − 1.79·43-s + 0.863·45-s − 0.723·47-s + 49-s − 0.344·55-s + 0.251·61-s + 3.55·63-s − 0.987·73-s − 1.41·77-s + 3·81-s + 3.95·83-s + 0.846·85-s + 0.863·95-s − 1.59·99-s + 3.99·101-s − 1.46·115-s + 3.48·119-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(6.980929586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.980929586\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 54 T - 12709 T^{2} - 54 p^{6} T^{3} + p^{12} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 610 T + 254451 T^{2} - 610 p^{6} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 1062 T - 643717 T^{2} + 1062 p^{6} T^{3} + p^{12} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9630 T + 68599331 T^{2} - 9630 p^{6} T^{3} + p^{12} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 20610 T + p^{6} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 142630 T + 14021953851 T^{2} + 142630 p^{6} T^{3} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 75150 T - 5131692829 T^{2} + 75150 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 57062 T - 48264302517 T^{2} - 57062 p^{6} T^{3} + p^{12} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 384050 T - 3839823789 T^{2} + 384050 p^{6} T^{3} + p^{12} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 1131030 T + p^{6} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.74751116846268358299634491900, −10.24570070772479838333031863155, −10.01162813067965740193671847386, −9.840689802201950905164466409692, −9.156691717379835533965203016662, −8.258658197847586147636029071413, −7.87533551881260985536524022590, −7.74581617009666701658670766252, −7.32296918580557159272536457028, −6.55532881516015027762061194481, −5.87302911062769265082332620743, −5.37153054498495259781864710926, −4.78964074962069973759460246322, −4.65788005593126897566070664373, −3.62243791744147743173623610779, −3.35920799227001125258870102661, −2.06951474335476487375749833703, −1.80929623559780080515127044967, −1.24484335328824013542240165516, −0.68362252089575950191616901191,
0.68362252089575950191616901191, 1.24484335328824013542240165516, 1.80929623559780080515127044967, 2.06951474335476487375749833703, 3.35920799227001125258870102661, 3.62243791744147743173623610779, 4.65788005593126897566070664373, 4.78964074962069973759460246322, 5.37153054498495259781864710926, 5.87302911062769265082332620743, 6.55532881516015027762061194481, 7.32296918580557159272536457028, 7.74581617009666701658670766252, 7.87533551881260985536524022590, 8.258658197847586147636029071413, 9.156691717379835533965203016662, 9.840689802201950905164466409692, 10.01162813067965740193671847386, 10.24570070772479838333031863155, 10.74751116846268358299634491900
